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I think I can try #1; #2 probably won't work since I have some 2.4GHz-only devices that need to connect to the extended network.

If #1 doesn't work try #2 temporarily. If that fixes the SBT then it's 2.4G interference from the extender. Try moving the extender somewhere else. I also didn't see in the extender manual if you could change the 2.4G channel the extender is using, but if you can try that too.

-Did you try rebooting the Touch when the range extender was on? Maybe the problem was that it was connected to the router, then saw the extender, and wasn't sure what to do.

-If the range extender is near the Touch, can you run an ethernet cable to connect the two? I now have an Orbi (as mentioned) and it's certainly a superior approach to range extenders. Many range extenders do have lightly documented or undocumented bridge capabilities too, which take no more to trigger than to simply run a ethernet cable to something you want to use it that way for. If the range extender does have an ethernet socket, try it. Remember to reconfigure your Touch to an ethernet connection rather than a wifi one.

Also take a look at ubiquiti unify products. Using them in and out the house and all connection problems are solved.

I'll second that. After years of intermittent wifi problems and poor coverage in some areas with a variety of consumer-grade routers, back in May this year I replaced my all-in-one router with a Ubiquiti EdgeRouter X ($38 with a Newegg promo) and a UAP-AC-LR Long Range Access Point ($95). I now have solid coverage throughout the house and all the way out to my detached garage about fifty feet away.